Problem: Is ${664632}$ divisible by $9$ ?
Solution: A number is divisible by $9$ if the sum of its digits is divisible by $9$ . [ Why? First, we can break the number up by place value: $ \begin{eqnarray} {664632}= &&{6}\cdot100000+ \\&&{6}\cdot10000+ \\&&{4}\cdot1000+ \\&&{6}\cdot100+ \\&&{3}\cdot10+ \\&&{2}\cdot1 \end{eqnarray} $ Next, we can rewrite each of the place values as $1$ plus a bunch of $9$ s: $ \begin{eqnarray} {664632}= &&{6}(99999+1)+ \\&&{6}(9999+1)+ \\&&{4}(999+1)+ \\&&{6}(99+1)+ \\&&{3}(9+1)+ \\&&{2} \end{eqnarray} $ Now if we distribute and rearrange, we get this: $ \begin{eqnarray} {664632}= &&\gray{6\cdot99999}+ \\&&\gray{6\cdot9999}+ \\&&\gray{4\cdot999}+ \\&&\gray{6\cdot99}+ \\&&\gray{3\cdot9}+ \\&& {6}+{6}+{4}+{6}+{3}+{2} \end{eqnarray} $ Any number consisting only of $9$ s is a multiple of $9$ , so the first five terms must all be multiples of $9$ That means that to figure out whether the original number is divisible by $9 $ , all we need to do is add up the digits and see if the sum is divisible by $9$ . In other words, ${664632}$ is divisible by $9$ if ${ 6}+{6}+{4}+{6}+{3}+{2}$ is divisible by $9$ Add the digits of ${664632}$ $ {6}+{6}+{4}+{6}+{3}+{2} = {27} $ If ${27}$ is divisible by $9$ , then ${664632}$ must also be divisible by $9$ ${27}$ is divisible by $9$, therefore ${664632}$ must also be divisible by $9$.